Probability/Random Variables CATALOG DESCRIPTION

Total Page：16

File Type：pdf, Size：1020Kb

ROWAN UNIVERSITY Department of Mathematics Syllabus STAT 02.360 - Probability/Random Variables CATALOG DESCRIPTION: STAT 02.360 Probability/Random Variables 3 s.h. (Prerequisites: Math 01.231 Calculus II and Math 01.150 Discrete Math with a grade of C- or better in both) An introduction to the theory and application of mathematical statistics at the post-calculus level. After a brief introduction to the concepts of descriptive statistics, the emphasis is on probability theory and its applications. Topics covered include sample spaces, random variables, discrete and continuous probability distributions, mathematical expectation, and multivariate distributions. Use of a graphing calculator is required. OBJECTIVES: Students will gain skills in using the theory of probability, combinatorics, and probability distributions to model applications. They will be able to recognize and apply many discrete and continuous probability distributions. They will derive and use moment generating functions, and study discrete and continuous random variables in enough depth to prepare them for the study of the Central Limit Theorem in Mathematical Statistics. CONTENT: 1. Descriptive Statistics 2. Elementary Probability 2.1 Axiomatic approach 2.2 Combinatorics 2.3 Probability theorems 21.4 Bayes' theorem 3. Discrete Probability Distributions 3.1 Discrete random variables 3.2 Expected values of a random variable and functions of a random variable 3.3 Binomial distribution 3.4 Geometric and negative binomial distributions 3.5 Hypergeometric distribution 3.6 Poisson distribution 3.7 Moment generating functions 4. Continuous Probability Distributions 4.1 Continuous random variables 4.2 Expected values of a random variable and functions of a random variable 4.3 Uniform distribution 4.4 Normal distribution 4.5 Gamma and exponential distributions 4.6 Beta distribution 4.7 Moment generating functions. 5. Multivariate Probability Distributions 5.1 Multivariate distributions 5.2 Marginal and conditional distributions 5.3 Independent random variables 5.4 Expected values for functions of random variables 5.5 Covariance and correlation 5.6 Linear Functions of random variables POSSIBLE TEXTS: Hastings, Probability and Statistics. Pearson, Boston, 1997. Hogg and Tannis, PROBABILITY AND STATISTICAL INFERENCE, 8th edition. Pearson, Boston, 2010. Larsen and Marx, An Introduction to Mathematical Statistics and Its Applications, 5th edition. Pearson, Boston, 2012. Mendenhall, Wackerly, and Schaeffer, MATHEMATICAL STATISTICS with APPLICATIONS, 7th edition, Brooks/Cole, Belmont, CA, 2008. .

Copy Link

Recommended publications

Mathematical Statistics
Mathematical Statistics MAS 713 Chapter 1.2 Previous subchapter 1 What is statistics ? 2 The statistical process 3 Population and samples 4 Random sampling Questions? Mathematical Statistics (MAS713) Ariel Neufeld 2 / 70 This subchapter Descriptive Statistics 1.2.1 Introduction 1.2.2 Types of variables 1.2.3 Graphical representations 1.2.4 Descriptive measures Mathematical Statistics (MAS713) Ariel Neufeld 3 / 70 1.2 Descriptive Statistics 1.2.1 Introduction Introduction As we said, statistics is the art of learning from data However, statistical data, obtained from surveys, experiments, or any series of measurements, are often so numerous that they are virtually useless unless they are condensed ; data should be presented in ways that facilitate their interpretation and subsequent analysis Naturally enough, the aspect of statistics which deals with organising, describing and summarising data is called descriptive statistics Mathematical Statistics (MAS713) Ariel Neufeld 4 / 70 1.2 Descriptive Statistics 1.2.2 Types of variables Types of variables Mathematical Statistics (MAS713) Ariel Neufeld 5 / 70 1.2 Descriptive Statistics 1.2.2 Types of variables Types of variables The range of available descriptive tools for a given data set depends on the type of the considered variable There are essentially two types of variables : 1 categorical (or qualitative) variables : take a value that is one of several possible categories (no numerical meaning) Ex. : gender, hair color, ﬁeld of study, political afﬁliation, status, ... 2 numerical (or quantitative)
[Show full text]
This History of Modern Mathematical Statistics Retraces Their Development
BOOK REVIEWS GORROOCHURN Prakash, 2016, Classic Topics on the History of Modern Mathematical Statistics: From Laplace to More Recent Times, Hoboken, NJ, John Wiley & Sons, Inc., 754 p. This history of modern mathematical statistics retraces their development from the “Laplacean revolution,” as the author so rightly calls it (though the beginnings are to be found in Bayes’ 1763 essay(1)), through the mid-twentieth century and Fisher’s major contribution. Up to the nineteenth century the book covers the same ground as Stigler’s history of statistics(2), though with notable differences (see infra). It then discusses developments through the first half of the twentieth century: Fisher’s synthesis but also the renewal of Bayesian methods, which implied a return to Laplace. Part I offers an in-depth, chronological account of Laplace’s approach to probability, with all the mathematical detail and deductions he drew from it. It begins with his first innovative articles and concludes with his philosophical synthesis showing that all fields of human knowledge are connected to the theory of probabilities. Here Gorrouchurn raises a problem that Stigler does not, that of induction (pp. 102-113), a notion that gives us a better understanding of probability according to Laplace. The term induction has two meanings, the first put forward by Bacon(3) in 1620, the second by Hume(4) in 1748. Gorroochurn discusses only the second. For Bacon, induction meant discovering the principles of a system by studying its properties through observation and experimentation. For Hume, induction was mere enumeration and could not lead to certainty. Laplace followed Bacon: “The surest method which can guide us in the search for truth, consists in rising by induction from phenomena to laws and from laws to forces”(5).
[Show full text]
Applied Time Series Analysis
Applied Time Series Analysis SS 2018 February 12, 2018 Dr. Marcel Dettling Institute for Data Analysis and Process Design Zurich University of Applied Sciences CH-8401 Winterthur Table of Contents 1 INTRODUCTION 1 1.1 PURPOSE 1 1.2 EXAMPLES 2 1.3 GOALS IN TIME SERIES ANALYSIS 8 2 MATHEMATICAL CONCEPTS 11 2.1 DEFINITION OF A TIME SERIES 11 2.2 STATIONARITY 11 2.3 TESTING STATIONARITY 13 3 TIME SERIES IN R 15 3.1 TIME SERIES CLASSES 15 3.2 DATES AND TIMES IN R 17 3.3 DATA IMPORT 21 4 DESCRIPTIVE ANALYSIS 23 4.1 VISUALIZATION 23 4.2 TRANSFORMATIONS 26 4.3 DECOMPOSITION 29 4.4 AUTOCORRELATION 50 4.5 PARTIAL AUTOCORRELATION 66 5 STATIONARY TIME SERIES MODELS 69 5.1 WHITE NOISE 69 5.2 ESTIMATING THE CONDITIONAL MEAN 70 5.3 AUTOREGRESSIVE MODELS 71 5.4 MOVING AVERAGE MODELS 85 5.5 ARMA(P,Q) MODELS 93 6 SARIMA AND GARCH MODELS 99 6.1 ARIMA MODELS 99 6.2 SARIMA MODELS 105 6.3 ARCH/GARCH MODELS 109 7 TIME SERIES REGRESSION 113 7.1 WHAT IS THE PROBLEM? 113 7.2 FINDING CORRELATED ERRORS 117 7.3 COCHRANE‐ORCUTT METHOD 124 7.4 GENERALIZED LEAST SQUARES 125 7.5 MISSING PREDICTOR VARIABLES 131 8 FORECASTING 137 8.1 STATIONARY TIME SERIES 138 8.2 SERIES WITH TREND AND SEASON 145 8.3 EXPONENTIAL SMOOTHING 152 9 MULTIVARIATE TIME SERIES ANALYSIS 161 9.1 PRACTICAL EXAMPLE 161 9.2 CROSS CORRELATION 165 9.3 PREWHITENING 168 9.4 TRANSFER FUNCTION MODELS 170 10 SPECTRAL ANALYSIS 175 10.1 DECOMPOSING IN THE FREQUENCY DOMAIN 175 10.2 THE SPECTRUM 179 10.3 REAL WORLD EXAMPLE 186 11 STATE SPACE MODELS 187 11.1 STATE SPACE FORMULATION 187 11.2 AR PROCESSES WITH MEASUREMENT NOISE 188 11.3 DYNAMIC LINEAR MODELS 191 ATSA 1 Introduction 1 Introduction 1.1 Purpose Time series data, i.e.
[Show full text]
Mathematical Statistics the Sample Distribution of the Median
Course Notes for Math 162: Mathematical Statistics The Sample Distribution of the Median Adam Merberg and Steven J. Miller February 15, 2008 Abstract We begin by introducing the concept of order statistics and ¯nding the density of the rth order statistic of a sample. We then consider the special case of the density of the median and provide some examples. We conclude with some appendices that describe some of the techniques and background used. Contents 1 Order Statistics 1 2 The Sample Distribution of the Median 2 3 Examples and Exercises 4 A The Multinomial Distribution 5 B Big-Oh Notation 6 C Proof That With High Probability jX~ ¡ ¹~j is Small 6 D Stirling's Approximation Formula for n! 7 E Review of the exponential function 7 1 Order Statistics Suppose that the random variables X1;X2;:::;Xn constitute a sample of size n from an in¯nite population with continuous density. Often it will be useful to reorder these random variables from smallest to largest. In reordering the variables, we will also rename them so that Y1 is a random variable whose value is the smallest of the Xi, Y2 is the next smallest, and th so on, with Yn the largest of the Xi. Yr is called the r order statistic of the sample. In considering order statistics, it is naturally convenient to know their probability density. We derive an expression for the distribution of the rth order statistic as in [MM]. Theorem 1.1. For a random sample of size n from an in¯nite population having values x and density f(x), the probability th density of the r order statistic Yr is given by ·Z ¸ ·Z ¸ n! yr r¡1 1 n¡r gr(yr) = f(x) dx f(yr) f(x) dx : (1.1) (r ¡ 1)!(n ¡ r)! ¡1 yr Proof.
[Show full text]
Nonparametric Multivariate Kurtosis and Tailweight Measures
Nonparametric Multivariate Kurtosis and Tailweight Measures Jin Wang1 Northern Arizona University and Robert Serfling2 University of Texas at Dallas November 2004 – final preprint version, to appear in Journal of Nonparametric Statistics, 2005 1Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona 86011-5717, USA. Email: [email protected]. 2Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75083- 0688, USA. Email: [email protected]. Website: www.utdallas.edu/∼serfling. Support by NSF Grant DMS-0103698 is gratefully acknowledged. Abstract For nonparametric exploration or description of a distribution, the treatment of location, spread, symmetry and skewness is followed by characterization of kurtosis. Classical moment- based kurtosis measures the dispersion of a distribution about its “shoulders”. Here we con- sider quantile-based kurtosis measures. These are robust, are defined more widely, and dis- criminate better among shapes. A univariate quantile-based kurtosis measure of Groeneveld and Meeden (1984) is extended to the multivariate case by representing it as a transform of a dispersion functional. A family of such kurtosis measures defined for a given distribution and taken together comprises a real-valued “kurtosis functional”, which has intuitive appeal as a convenient two-dimensional curve for description of the kurtosis of the distribution. Several multivariate distributions in any dimension may thus be compared with respect to their kurtosis in a single two-dimensional plot. Important properties of the new multivariate kurtosis measures are established. For example, for elliptically symmetric distributions, this measure determines the distribution within affine equivalence. Related tailweight measures, influence curves, and asymptotic behavior of sample versions are also discussed.
[Show full text]
Why Distinguish Between Statistics and Mathematical Statistics – the Case of Swedish Academia
Why distinguish between statistics and mathematical statistics { the case of Swedish academia Peter Guttorp1 and Georg Lindgren2 1Department of Statistics, University of Washington, Seattle 2Mathematical Statistics, Lund University, Lund 2017/10/11 Abstract A separation between the academic subjects statistics and mathematical statistics has existed in Sweden almost as long as there have been statistics professors. The same distinction has not been maintained in other countries. Why has it been kept so for long in Sweden, and what consequences may it have had? In May 2015 it was 100 years since Mathematical Statistics was formally estab- lished as an academic discipline at a Swedish university where Statistics had existed since the turn of the century. We give an account of the debate in Lund and elsewhere about this division dur- ing the first decades after 1900 and present two of its leading personalities. The Lund University astronomer (and mathematical statistician) C.V.L. Charlier was a lead- ing proponent for a position in mathematical statistics at the university. Charlier's adversary in the debate was Pontus Fahlbeck, professor in political science and statis- tics, who reserved the word statistics for \statistics as a social science". Charlier not only secured the first academic position in Sweden in mathematical statistics for his former Ph.D. student Sven Wicksell, but he also demonstrated that a mathematical statistician can be influential in matters of state, finance, as well as in different nat- ural sciences. Fahlbeck saw mathematical statistics as a set of tools that sometimes could be useful in his brand of statistics. After a summary of the organisational, educational, and scientific growth of the statistical sciences in Sweden that has taken place during the last 50 years, we discuss what effects the Charlier-Fahlbeck divergence might have had on this development.
[Show full text]
Mathematical Expectation
Mathematical Expectation 1 Introduction 6 Product Moments 2 The Expected Value of a Random 7 Moments of Linear Combinations of Random Variable Variables 3 Moments 8 Conditional Expectations 4 Chebyshev’s Theorem 9 The Theory in Practice 5 Moment-Generating Functions 1 Introduction Originally, the concept of a mathematical expectation arose in connection with games of chance, and in its simplest form it is the product of the amount a player stands to win and the probability that he or she will win. For instance, if we hold one of 10,000 tickets in a rafﬂe for which the grand prize is a trip worth $4,800, our mathemati- cal expectation is 4,800 1 $0 .48. This amount will have to be interpreted in · 10,000 = the sense of an average—altogether the 10,000 tickets pay $4,800, or on the average $4,800 $0 .48 per ticket. 10,000 = If there is also a second prize worth $1,200 and a third prize worth $400, we can argue that altogether the 10,000 tickets pay $4,800 $1,200 $400 $6,400, or on + + = the average $6,400 $0 .64 per ticket. Looking at this in a different way, we could 10,000 = argue that if the rafﬂe is repeated many times, we would lose 99.97 percent of the time (or with probability 0.9997) and win each of the prizes 0.01 percent of the time (or with probability 0.0001). On the average we would thus win 0(0.9997 ) 4,800 (0.0001 ) 1,200 (0.0001 ) 400 (0.0001 ) $0 .64 + + + = which is the sum of the products obtained by multiplying each amount by the corre- sponding probability.
[Show full text]
Chapter 5: Multivariate Distributions
Chapter 5: Multivariate Distributions Professor Ron Fricker Naval Postgraduate School Monterey, California 3/15/15 Reading Assignment: Sections 5.1 – 5.12 1 Goals for this Chapter • Bivariate and multivariate probability distributions – Bivariate normal distribution – Multinomial distribution • Marginal and conditional distributions • Independence, covariance and correlation • Expected value & variance of a function of r.v.s – Linear functions of r.v.s in particular • Conditional expectation and variance 3/15/15 2 Section 5.1: Bivariate and Multivariate Distributions • A bivariate distribution is a probability distribution on two random variables – I.e., it gives the probability on the simultaneous outcome of the random variables • For example, when playing craps a gambler might want to know the probability that in the simultaneous roll of two dice each comes up 1 • Another example: In an air attack on a bunker, the probability the bunker is not hardened and does not have SAM protection is of interest • A multivariate distribution is a probability distribution for more than two r.v.s 3/15/15 3 Joint Probabilities • Bivariate and multivariate distributions are joint probabilities – the probability that two or more events occur – It’s the probability of the intersection of n 2 ≥ events: Y = y , Y = y ,..., Y = y { 1 1} { 2 2} { n n} – We’ll denote the joint (discrete) probabilities as P (Y1 = y1,Y2 = y2,...,Yn = yn) • We’ll sometimes use the shorthand notation p(y1,y2,...,yn) – This is the probability that the event Y = y and { 1 1} the
[Show full text]
Mathematical Statistics in 20Th-Century America
HISTORIA MATHEMATICA 23 (1996), 7±30 ARTICLE NO. 0002 Drawing the Boundaries: Mathematical Statistics in 20th-Century America PATTI W. HUNTER View metadata, citation and similar papers at core.ac.uk brought to you by CORE Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903-3199 provided by Elsevier - Publisher Connector Statistics in Europe and in the United States acquired the status of a separate ®eld only in the 20th century. Until then, it had provided a quantitative tool for analyzing data in various sciences. In the early 1930s, the founding of the Annals of Mathematical Statistics and of the Institute of Mathematical Statistics served to institutionalize the profession of mathematical statistics in the United States, and helped to distinguish it as a discipline independent from the social sciences which had previously supported it. This paper examines the American mathematical statisticians' attempts to delineate the boundaries of their profession, focusing on their association with the social sciences, on the founding of the Annals and of the IMS, and on the profession's continuing advancement during World War II. 1996 Academic Press, Inc. Il n'eÂtait qu'au 20eÁme sieÁcle que la statistique est devenue une discipline distincte. AnteÂrieure- ment, elle a servi d'outil quantitatif pour l'analyse des donneÂes dans les sciences diverses. Dans les 1930s, l'eÂtablissement des Annals of Mathematical Statistics et de l'Institute of Mathematical Statistics a ameneÂ l'institutionalisation d'une profession ``statistique matheÂmatique'' et a servi aÁ distinguer cette profession des sciences sociales. Ici, j'analyse les tentatives des statisticiens matheÂmatiques ameÂricains d'eÂtablir les frontieÁres de leur discipline en concentrant sur leurs associations avec les sciences sociales, le fondement des Annals et de l'IMS, et le progreÁsde la profession pendant la deuxieÁme guerre mondiale.
[Show full text]
Applying Generalized Linear Models
Applying Generalized Linear Models James K. Lindsey Springer Preface Generalized linear models provide a unified approach to many of the most common statistical procedures used in applied statistics. They have ap- plications in disciplines as widely varied as agriculture, demography, eco- logy, economics, education, engineering, environmental studies and pollu- tion, geography, geology, history, medicine, political science, psychology, and sociology, all of which are represented in this text. In the years since the term was first introduced by Nelder and Wedder- burn in 1972, generalized linear models have slowly become well known and widely used. Nevertheless, introductory statistics textbooks, and courses, still most often concentrate on the normal linear model, just as they did in the 1950s, as if nothing had happened in statistics in between. For students who will only receive one statistics course in their career, this is especially disastrous, because they will have a very restricted view of the possible utility of statistics in their chosen field of work. The present text, being fairly advanced, is not meant to fill that gap; see, rather, Lindsey (1995a). Thus, throughout much of the history of statistics, statistical modelling centred around this normal linear model. Books on this subject abound. More recently, log linear and logistic models for discrete, categorical data have become common under the impetus of applications in the social sci- ences and medicine. A third area, models for survival data, also became a growth industry, although not always so closely related to generalized linear models. In contrast, relatively few books on generalized linear models, as such, are available.
[Show full text]
A General Approach to Teaching Random Variables
1 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 A GENERAL APPROACH TO TEACHING RANDOM VARIABLES Farida Kachapova School of Computing and Mathematical Sciences Auckland University of Technology Private Bag 92006 Auckland 1142 New Zealand Email: [email protected] Abstract The paper describes a teaching approach in post-calculus probability courses when initial topics on random variables are taught in general form instead of doing it separately for discrete and continuous cases. Some common student misconceptions about random variables are analyzed and accompanied with counter-examples. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj 2 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 INTRODUCTION Most research on teaching probability focus on basic probability and statistical applications. The teaching of probability concepts, including random variables, is studied by Larsen (2006), Miller (1998), Leviatan (2003), Ballman (1997), and Hernandez, Albert Huerta and Batanero (2006). Much research has been done on students’ thinking about probability and related misconceptions (see, for example, Ancker, 2006; Rossman, 1995; San Martin, 2006; Warner, Pendergraft, and Webb, 1998). Random variable is “... a fundamental stochastic idea” and “..
[Show full text]
Theory of Probability and Mathematical Statistics
Theory of probability and mathematical statistics TomáˇsMrkviˇcka Bibliography [1] J. Andd_z}l:ˇ Matematickd_z}statistikaˇ , SNTL/ALFA, Praha 1978 [2] J. Andd_z}l:ˇ Statistickd_z}metodyˇ , Matfyzpress, Praha 1998 [3] V. Dupad_z},M.ˇ Hud_z}kovˇ d_z}:ˇ Pravdd_z}podobnostˇ a matematickd_z}statis-ˇ tika, Karolinum, Praha 1999 [4] T. Mrkvid_z}ka,ˇ V. Petrd_z}ˇ d_z}kovˇ d_z}:ˇ d_z}vodˇ do teorie pravdd_z}podobnostiˇ ,PF JU, d_z}eskˇ d_z}Budˇ d_z}joviceˇ 2008 [5] T. Mrkvid_z}ka,ˇ V. Petrd_z}ˇ d_z}kovˇ d_z}:ˇ d_z}vodˇ do statistiky,PF JU, d_z}eskˇ d_z}ˇ Budd_z}joviceˇ 2006 1 Chapter 1 Probability theory 2 CHAPTER 1. PROBABILITY THEORY 3 1.1 Definition of probability Definition 1.1 Classical definition: Let A1, ... , An be random events, such that • every time one and only one random event happen, • all the event are equally probable. And let the event A happen if happen one of the event Ai1, ... , Aik . Then the probability of A is P (A) = k=n. 1.1.1 Kolmogorov definition of probability Probability space (Ω; A;P ). Ω is nonempty set of all results of the random experiment, results are !. A is σ-algebra on Ω. (The set of all "nice" subsets of Ω) P : A! R is function giving every set A 2 A its probability (0 ≤ P (A) ≤ 1, P (Ω) = 1). This is probability measure. CHAPTER 1. PROBABILITY THEORY 4 Rules P (AC) = 1 − P (A) P (A [ B) = P (A) + P (B); if A, B are disjoint P (A [ B) = P (A) + P (B) − P (A \ B) CHAPTER 1.
[Show full text]